# Source code for statsmodels.stats.mediation

```"""
Mediation analysis

Implements algorithm 1 ('parametric inference') and algorithm 2
('nonparametric inference') from:

Imai, Keele, Tingley (2010).  A general approach to causal mediation
analysis. Psychological Methods 15:4, 309-334.

http://imai.princeton.edu/research/files/BaronKenny.pdf

The algorithms are described on page 317 of the paper.

In the case of linear models with no interactions involving the
mediator, the results should be similar or identical to the earlier
Barron-Kenny approach.
"""

import numpy as np
import pandas as pd
from statsmodels.graphics.utils import maybe_name_or_idx
import statsmodels.compat.pandas as pdc  # pragma: no cover

[docs]class Mediation(object):
"""
Conduct a mediation analysis.

Parameters
----------
outcome_model : statsmodels model
Regression model for the outcome.  Predictor variables include
the treatment/exposure, the mediator, and any other variables
of interest.
mediator_model : statsmodels model
Regression model for the mediator variable.  Predictor
variables include the treatment/exposure and any other
variables of interest.
exposure : string or (int, int) tuple
The name or column position of the treatment/exposure
variable.  If positions are given, the first integer is the
column position of the exposure variable in the outcome model
and the second integer is the position of the exposure variable
in the mediator model.  If a string is given, it must be the name
of the exposure variable in both regression models.
mediator : string or int
The name or column position of the mediator variable in the
outcome regression model.  If None, infer the name from the
mediator model formula (if present).
moderators : dict
Map from variable names or index positions to values of
moderator variables that are held fixed when calculating
mediation effects.  If the keys are index position they must
be tuples `(i, j)` where `i` is the index in the outcome model
and `j` is the index in the mediator model.  Otherwise the
keys must be variable names.
outcome_fit_kwargs : dict-like
Keyword arguments to use when fitting the outcome model.
mediator_fit_kwargs : dict-like
Keyword arguments to use when fitting the mediator model.

Returns a ``MediationResults`` object.

Notes
-----
The mediator model class must implement ``get_distribution``.

Examples
--------
A basic mediation analysis using formulas:

>>> import statsmodels.api as sm
>>> outcome_model = sm.GLM.from_formula("cong_mesg ~ emo + treat + age + educ + gender + income",
>>> mediator_model = sm.OLS.from_formula("emo ~ treat + age + educ + gender + income", data)
>>> med = Mediation(outcome_model, mediator_model, "treat", "emo").fit()
>>> med.summary()

A basic mediation analysis without formulas.  This may be slightly
faster than the approach using formulas.  If there are any
interactions involving the treatment or mediator variables this
approach will not work, you must use formulas.

>>> import patsy
>>> outcome = np.asarray(data["cong_mesg"])
>>> outcome_exog = patsy.dmatrix("emo + treat + age + educ + gender + income", data,
...                              return_type='dataframe')
>>> outcome_model = sm.GLM(outcome, outcome_exog, family=sm.families.Binomial(link=probit))
>>> mediator = np.asarray(data["emo"])
>>> mediator_exog = patsy.dmatrix("treat + age + educ + gender + income", data,
...                               return_type='dataframe')
>>> mediator_model = sm.OLS(mediator, mediator_exog)
>>> tx_pos = [outcome_exog.columns.tolist().index("treat"),
...           mediator_exog.columns.tolist().index("treat")]
>>> med_pos = outcome_exog.columns.tolist().index("emo")
>>> med = Mediation(outcome_model, mediator_model, tx_pos, med_pos).fit()
>>> med.summary()

A moderated mediation analysis.  The mediation effect is computed
for people of age 20.

>>> fml = "cong_mesg ~ emo + treat*age + emo*age + educ + gender + income",
>>> outcome_model = sm.GLM.from_formula(fml, data,
...                                      family=sm.families.Binomial())
>>> mediator_model = sm.OLS.from_formula("emo ~ treat*age + educ + gender + income", data)
>>> moderators = {"age" : 20}
>>> med = Mediation(outcome_model, mediator_model, "treat", "emo",
...                 moderators=moderators).fit()

References
----------
Imai, Keele, Tingley (2010).  A general approach to causal mediation
analysis. Psychological Methods 15:4, 309-334.
http://imai.princeton.edu/research/files/BaronKenny.pdf

Tingley, Yamamoto, Hirose, Keele, Imai (2014).  mediation : R
package for causal mediation analysis.  Journal of Statistical
Software 59:5.  http://www.jstatsoft.org/v59/i05/paper
"""

def __init__(self, outcome_model, mediator_model, exposure, mediator=None,
moderators=None, outcome_fit_kwargs=None, mediator_fit_kwargs=None):

self.outcome_model = outcome_model
self.mediator_model = mediator_model
self.exposure = exposure
self.moderators = moderators if moderators is not None else {}

if mediator is None:
self.mediator = self._guess_endog_name(mediator_model, 'mediator')
else:
self.mediator = mediator

self._outcome_fit_kwargs = (outcome_fit_kwargs if outcome_fit_kwargs
is not None else {})
self._mediator_fit_kwargs = (mediator_fit_kwargs if mediator_fit_kwargs
is not None else {})

# We will be changing these so need to copy.
self._outcome_exog = outcome_model.exog.copy()
self._mediator_exog = mediator_model.exog.copy()

# Position of the exposure variable in the mediator model.
self._exp_pos_mediator = self._variable_pos('exposure', 'mediator')

# Position of the exposure variable in the outcome model.
self._exp_pos_outcome = self._variable_pos('exposure', 'outcome')

# Position of the mediator variable in the outcome model.
self._med_pos_outcome = self._variable_pos('mediator', 'outcome')

def _variable_pos(self, var, model):
if model == 'mediator':
mod = self.mediator_model
else:
mod = self.outcome_model

if var == 'mediator':
return maybe_name_or_idx(self.mediator, mod)[1]

exp = self.exposure
exp_is_2 = ((len(exp) == 2) and (type(exp) != type('')))

if exp_is_2:
if model == 'outcome':
return exp[0]
elif model == 'mediator':
return exp[1]
else:
return maybe_name_or_idx(exp, mod)[1]

def _guess_endog_name(self, model, typ):
if hasattr(model, 'formula'):
return model.formula.split("~")[0].strip()
else:
raise ValueError('cannot infer %s name without formula' % typ)

def _simulate_params(self, result):
"""
Simulate model parameters from fitted sampling distribution.
"""
mn = result.params
cov = result.cov_params()
return np.random.multivariate_normal(mn, cov)

def _get_mediator_exog(self, exposure):
"""
Return the mediator exog matrix with exposure set to the given
value.  Set values of moderated variables as needed.
"""
mediator_exog = self._mediator_exog
if not hasattr(self.mediator_model, 'formula'):
mediator_exog[:, self._exp_pos_mediator] = exposure
for ix in self.moderators:
v = self.moderators[ix]
mediator_exog[:, ix[1]] = v
else:
# Need to regenerate the model exog
df = self.mediator_model.data.frame.copy()
df.loc[:, self.exposure] = exposure
for vname in self.moderators:
v = self.moderators[vname]
df.loc[:, vname] = v
klass = self.mediator_model.__class__
init_kwargs = self.mediator_model._get_init_kwds()
model = klass.from_formula(data=df, **init_kwargs)
mediator_exog = model.exog

return mediator_exog

def _get_outcome_exog(self, exposure, mediator):
"""
Retun the exog design matrix with mediator and exposure set to
the given values.  Set values of moderated variables as
needed.
"""
outcome_exog = self._outcome_exog
if not hasattr(self.outcome_model, 'formula'):
outcome_exog[:, self._med_pos_outcome] = mediator
outcome_exog[:, self._exp_pos_outcome] = exposure
for ix in self.moderators:
v = self.moderators[ix]
outcome_exog[:, ix[0]] = v
else:
# Need to regenerate the model exog
df = self.outcome_model.data.frame.copy()
df.loc[:, self.exposure] = exposure
df.loc[:, self.mediator] = mediator
for vname in self.moderators:
v = self.moderators[vname]
df.loc[:, vname] = v
klass = self.outcome_model.__class__
init_kwargs = self.outcome_model._get_init_kwds()
model = klass.from_formula(data=df, **init_kwargs)
outcome_exog = model.exog

return outcome_exog

def _fit_model(self, model, fit_kwargs, boot=False):
klass = model.__class__
init_kwargs = model._get_init_kwds()
endog = model.endog
exog = model.exog
if boot:
ii = np.random.randint(0, len(endog), len(endog))
endog = endog[ii]
exog = exog[ii, :]
outcome_model = klass(endog, exog, **init_kwargs)
return outcome_model.fit(**fit_kwargs)

[docs]    def fit(self, method="parametric", n_rep=1000):
"""
Fit a regression model to assess mediation.

Parameters
----------
method : string
Either 'parametric' or 'bootstrap'.
n_rep : integer
The number of simulation replications.

Returns a MediationResults object.
"""

if method.startswith("para"):
# Initial fit to unperturbed data.
outcome_result = self._fit_model(self.outcome_model, self._outcome_fit_kwargs)
mediator_result = self._fit_model(self.mediator_model, self._mediator_fit_kwargs)
elif not method.startswith("boot"):
raise("method must be either 'parametric' or 'bootstrap'")

indirect_effects = [[], []]
direct_effects = [[], []]

for iter in range(n_rep):

if method == "parametric":
# Realization of outcome model parameters from sampling distribution
outcome_params = self._simulate_params(outcome_result)

# Realization of mediation model parameters from sampling distribution
mediation_params = self._simulate_params(mediator_result)
else:
outcome_result = self._fit_model(self.outcome_model,
self._outcome_fit_kwargs, boot=True)
outcome_params = outcome_result.params
mediator_result = self._fit_model(self.mediator_model,
self._mediator_fit_kwargs, boot=True)
mediation_params = mediator_result.params

# predicted outcomes[tm][te] is the outcome when the
# mediator is set to tm and the outcome/exposure is set to
# te.
predicted_outcomes = [[None, None], [None, None]]
for tm in 0, 1:
mex = self._get_mediator_exog(tm)
gen = self.mediator_model.get_distribution(mediation_params,
mediator_result.scale,
exog=mex)
potential_mediator = gen.rvs(mex.shape[0])

for te in 0, 1:
oex = self._get_outcome_exog(te, potential_mediator)
po = self.outcome_model.predict(outcome_params, oex)
predicted_outcomes[tm][te] = po

for t in 0, 1:
indirect_effects[t].append(predicted_outcomes[1][t] - predicted_outcomes[0][t])
direct_effects[t].append(predicted_outcomes[t][1] - predicted_outcomes[t][0])

for t in 0, 1:
indirect_effects[t] = np.asarray(indirect_effects[t]).T
direct_effects[t] = np.asarray(direct_effects[t]).T

self.indirect_effects = indirect_effects
self.direct_effects = direct_effects

rslt = MediationResults(self.indirect_effects, self.direct_effects)
rslt.method = method
return rslt

def _pvalue(vec):
return 2 * min(sum(vec > 0), sum(vec < 0)) / float(len(vec))

[docs]class MediationResults(object):
"""
A class for holding the results of a mediation analysis.

The following terms are used in the summary output:

ACME : average causal mediated effect
"""

def __init__(self, indirect_effects, direct_effects):

self.indirect_effects = indirect_effects
self.direct_effects = direct_effects

indirect_effects_avg = [None, None]
direct_effects_avg = [None, None]
for t in 0, 1:
indirect_effects_avg[t] = indirect_effects[t].mean(0)
direct_effects_avg[t] = direct_effects[t].mean(0)

self.ACME_ctrl = indirect_effects_avg[0]
self.ACME_tx = indirect_effects_avg[1]

self.prop_med_ctrl = self.ACME_ctrl / self.total_effect
self.prop_med_tx = self.ACME_tx / self.total_effect
self.prop_med_avg = (self.prop_med_ctrl + self.prop_med_tx) / 2

self.ACME_avg = (self.ACME_ctrl + self.ACME_tx) / 2

[docs]    def summary(self, alpha=0.05):
"""
Provide a summary of a mediation analysis.
"""

columns = ["Estimate", "Lower CI bound", "Upper CI bound", "P-value"]
"Total effect", "Prop. mediated (control)", "Prop. mediated (treated)",
"ACME (average)", "ADE (average)", "Prop. mediated (average)"]
smry = pd.DataFrame(columns=columns, index=index)

self.total_effect, self.prop_med_ctrl,
self.prop_med_avg]):

if ((vec is self.prop_med_ctrl) or (vec is self.prop_med_tx) or
(vec is self.prop_med_avg)):
smry.iloc[i, 0] = np.median(vec)
else:
smry.iloc[i, 0] = vec.mean()
smry.iloc[i, 1] = np.percentile(vec, 100 * alpha / 2)
smry.iloc[i, 2] = np.percentile(vec, 100 * (1 - alpha / 2))
smry.iloc[i, 3] = _pvalue(vec)

if pdc.version < '0.17.0':  # pragma: no cover
smry = smry.convert_objects(convert_numeric=True)
else:  # pragma: no cover
smry = smry.apply(pd.to_numeric, errors='coerce')

return smry
```